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Feb 13, 2023 24 likes •288 views

Base Tangle Decompositions and Subdivisions of Knots and Links By M. Ochiai, N.Morimura, and A.Emoto HOMFLY invariant of Knot or Link For a knot or link K, (1)P(K;x,y) = 1 if K is the trivial knot (2) x P(K+;x,y) + y P(K-;x,y) = P(K

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Base Tangle Decompositions and Subdivisions of Knots and Links

By M. Ochiai, N.Morimura, and A.Emoto

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HOMFLY invariant of Knot or Link For a knot or link K, (1)P(K;x,y) = 1 if K is the trivial knot (2) x P(K+;x,y) + y P(K-;x,y) = P(K∞;x,y)

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Let T be a 2-string tangle. x P(T+;x,y) + y P(T-;x,y) = P(T ∞ ;x,y) K = T1 + T2, wheren K is a knot and T1 ,T2 are tangles. P(T1; x,y) = a1 A1 + b1 B1 P(T2; x,y) = a2 A2 + b2 B2, P(K;x,y)=a1a2(A1+A2)+a1b2(A1+B2)+ b1a2(B1+A2)+b1b2(B1+B2) Where A1, B1, A2, B2 are base tangles. A tangle T is called by a base tangle if (1) T has minimal crossings with only plus sign.

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Tangle decomposition of Conway’s knot

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n-tangle decomposition(n>2)

Base tangle by using oriented ordered

tangle (1) Minimal crossings (2) First string has only over (resp. under) crossings, second string has also over (resp. under) crossings other than crossing with the first string,….

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Reconstruction of a base tangle from a sequence

Our first method uses to make a base tangle from a

sequence by using small deformations of string end points (The last version).

The second method makes to classify n-multiple

points which have models of n double points. This program made by Miss A. Emoto. And this program can make n string base tangles with up to

12. But practically, up to 9-string, in this case

there exists 9! base tangles.

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The base tangle decomposition of the first tangle of Terasaka’s knot

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The base tangle decomposition of the second tangle of Terasaka’s knot

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The 3-parallel HOMFLY polynomial of Terasaka’s knot

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Final computation of base tangle decomposition

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How to compute the 3-parallel HOMFLY polynomial of the following knot with 15-crossings

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Obstruction of subdivision

Usual base decompositions are applied to 2

different strings and so no “free loops” are generated.

Tangle subdivision generates at most three

free loops.

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Subdivision programs

composition(0, 12, 12, 12, 0, 0, NULL); >> knot , link composition(1, 12, 18, 9, p, q, base); composition 696() >> 9-string composition(2, 12, 18, 6, p, q, base); composition699() >> 6-string composition(3, 12, 12, 6, p, q, base); composition666() >> 6-string composition(4, 18, 18, 6, p, q, base); composition9912() >> 6-string